The continuum fallacy (also called the fallacy of the beard,[1]line drawing fallacy, bald man fallacy, fallacy of the heap, the sorites fallacy) is an informal fallacy closely related to the sorites paradox, or paradox of the heap. The fallacy causes one to erroneously reject a vagueclaim simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity.

The fallacy is the argument that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states.

There are clearly reasonable and clearly unreasonable cases in which objects either belong or do not belong to a particular group of objects based on their properties. We are able to take them case by case and designate them as such even in the case of properties which may be vaguely defined. The existence of hard or controversial cases does not preclude our ability to designate members of particular kinds of groups.

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Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.

For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample: there do exist bald people and people who aren't bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these "slightly"s build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough "slightly"s will certify the group's heap status – see fuzzy logic.

Imagine grains of sand in a bag. I can lift the bag when it contains one grain of sand. If I can lift the bag with N grains of sand then I can certainly lift it with N+1 grains of sand (for it is absurd to think I can lift N grains but adding a single grain makes it too heavy to lift). Therefore, I can lift the bag when it has any number of grains of sand, even if it has five tons of sand.